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<b>let </b><font color="Maroon">x</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<a NAME="E1:66"/><i><font color="Green">E21</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="square_1.html#K3">^2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 &amp;  <a href="square_1.html#K6">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
<b>by </b><i>, , , <a class="txt" href="sin_cos7.html#E21"><i><font color="Green">Lemma42</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">t</font> = <span class="p1">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x</font>;<br/>
 <a href="sin_cos7.html#K4" target="_self">tanh"</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="power.html#K5">log</a> <a href="power.html#K8">number_e</a> ,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T114">XCMPLX_1:114</a>, <a class="ref" href="sin_cos7.html#T30" target="_self">Th30</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="power.html#K5">log</a> <a href="power.html#K8">number_e</a> ,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K3">*</a> <span class="p0">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T128">XCMPLX_1:128</a>, <a class="ref" href="sin_cos7.html#T30" target="_self">Th30</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="power.html#K5">log</a> <a href="power.html#K8">number_e</a> ,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T55">XCMPLX_1:55</a>, <a class="ref" href="sin_cos7.html#T30" target="_self">Th30</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="power.html#K5">log</a> <a href="power.html#K8">number_e</a> ,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T92">XCMPLX_1:92</a>, <a class="ref" href="sin_cos7.html#T30" target="_self">Th30</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="power.html#K5">log</a> <a href="power.html#K8">number_e</a> ,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="power.html#K5">log</a> <a href="power.html#K8">number_e</a> ,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos7.html#T30" target="_self">Th30</a>, <a class="ref" href="square_1.html#T97">SQUARE_1:97</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="power.html#K5">log</a> <a href="power.html#K8">number_e</a> ,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="square_1.html#T89">SQUARE_1:89</a>, <a class="ref" href="sin_cos7.html#T30" target="_self">Th30</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="power.html#K5">log</a> <a href="power.html#K8">number_e</a> ,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x</font></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="power.html#K5">log</a> <a href="power.html#K8">number_e</a> ,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x</font></span>)</span> <a href="power.html#K3">to_power</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T53">POWER:53</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="power.html#K5">log</a> <a href="power.html#K8">number_e</a> ,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T63">POWER:63</a>, <a class="ref" href="sin_cos7.html#T30" target="_self">Th30</a>, </i>
<br/>.= 
 <a href="power.html#K5">log</a> <a href="power.html#K8">number_e</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x</font></span>)</span>

;<br/>
<b>hence </b><a NAME="E3:66"/>
 <a href="sin_cos7.html#K1" target="_self">sinh"</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="sin_cos7.html#K4" target="_self">tanh"</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span>
 ;<br/>


</div>
